The sides of a triangle are in the ratio 4:6:7. Then

• A. The triangle is obtuse-angled
• B. The triangle is acute-angled
• C. The triangle is right-angled
• D. The triangle is impossible

The sides of a triangle are given in the ratio 4:6:7. To determine the type of triangle, we can use the properties of triangle sides and the Pythagorean theorem.

Understanding Triangle Types

Triangles can be classified based on their angles:

• Acute-angled: All angles are less than 90 degrees.
• Right-angled: One angle is exactly 90 degrees.
• Obtuse-angled: One angle is greater than 90 degrees.

Calculating the Sides

Let the sides of the triangle be represented as:

• 4x
• 6x
• 7x

To check the type of triangle, we can apply the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side.

Applying the Triangle Inequality

We need to check the following conditions:

• 4x + 6x > 7x
• 4x + 7x > 6x
• 6x + 7x > 4x

Calculating these:

• 10x > 7x (True)
• 11x > 6x (True)
• 13x > 4x (True)

Since all conditions are satisfied, a triangle can be formed.

Determining the Angle Type

Next, we can check if the triangle is obtuse, acute, or right by comparing the squares of the sides:

• 4x: (4x)² = 16x²
• 6x: (6x)² = 36x²
• 7x: (7x)² = 49x²

Now, we check if:

• 16x² + 36x² < 49x² (for obtuse)
• 16x² + 36x² = 49x² (for right)
• 16x² + 36x² > 49x² (for acute)

Calculating:

16x² + 36x² = 52x², which is less than 49x². Thus, the triangle is obtuse-angled.

Final Answer

The correct option is A. The triangle is obtuse-angled.